This review elaborates pedagogically on the fundamental concept, basic theory, expected properties, and materials realizations of topological superconductors. The relation between topological superconductivity and Majorana fermions are explained, and the difference between dispersive Majorana fermions and a localized Majorana zero mode is emphasized. A variety of routes to topological superconductivity are explained with an emphasis on the roles of spin–orbit coupling. Present experimental situations and possible signatures of topological superconductivity are summarized with an emphasis on intrinsic topological superconductors.

Topological insulators represent a new quantum state of matter which is characterized by peculiar edge or surface states that show up due to a topological character of the bulk wave functions. This review presents a pedagogical account on topological insulator materials with an emphasis on basic theory and materials properties. After presenting a historical perspective and basic theories of topological insulators, it discusses all the topological insulator materials discovered as of May 2013, with some illustrative descriptions of the developments in materials discoveries in which the author was involved. A summary is given for possible ways to confirm the topological nature in a candidate material. Various synthesis techniques as well as the defect chemistry that are important for realizing bulk-insulating samples are discussed. Characteristic properties of topological insulators are discussed with an emphasis on transport properties. In particular, the Dirac fermion physics and the resulting peculiar quantum oscillation patterns are discussed in detail. It is emphasized that proper analyses of quantum oscillations make it possible to unambiguously identify surface Dirac fermions through transport measurements. The prospects of topological insulator materials for elucidating novel quantum phenomena that await discovery conclude the review.Comment: 36 pages, 19 figures, 300 references; invited review paper published in J. Phys. Soc. Jpn. (free access); final version with further error correction

In this review, we discuss recent progress in the explorations of topological materials beyond topological insulators; specifically, we focus on topological crystalline insulators and bulk topological superconductors. The basic concepts, model Hamiltonians, and novel electronic properties of these new topological materials are explained. The key role of the symmetries that underlie their topological properties is elucidated. Key issues in their materials realizations are also discussed.

A topological superconductor (TSC) is characterized by the topologically protected gapless surface state that is essentially an Andreev bound state consisting of Majorana fermions. While a TSC has not yet been discovered, the doped topological insulator ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$, which superconducts below $\sim 3\mathrm{K}$, has been predicted to possess a topological superconducting state. We report that the point-contact spectra on the cleaved surface of superconducting ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$ present a zero-bias conductance peak (ZBCP) which signifies unconventional superconductivity. Theoretical considerations of all possible superconducting states help us conclude that this ZBCP is due to Majorana fermions and gives evidence for a topological superconductivity in ${\mathrm{Cu}}_x{\mathrm{Bi}}_2{\mathrm{Se}}_3$. In addition, we found an unusual pseudogap that develops below $\sim 20\mathrm{K}$ and coexists with the topological superconducting state.

Topological insulators are predicted to present interesting surface transport phenomena but their experimental studies have been hindered by a metallic bulk conduction that overwhelms the surface transport. We show that the topological insulator Bi2Te2Se presents a high resistivity exceeding 1$\hskip0.3em${}$\Omega${}\,cm and a variable-range hopping behavior, and yet presents Shubnikov-de Haas oscillations coming from the topological surface state. Furthermore, we have been able to clarify both the bulk and surface transport channels, establishing a comprehensive understanding of the transport in this material. Our results demonstrate that Bi2Te2Se is, to our knowledge, the best material to date for studying the surface quantum transport in a topological insulator.